I noticed that the nice symmetric layout of a grid graph gets distorted when additional edges are added to the graph. Is there way to prevent that from happening and to keep the rectangular, grid-like layout?

I would like the additional edge(s) added to the original graph, keeping its layout unchanged.

Thanks.

Edit:
I realize that

EdgeAdd[g, 1 \[UndirectedEdge] 5]

does what I want. I'd still be interested to learn how I can force a grid-like layout when constructing the graph explicitly from vertex and edge lists, in particular since EdgeAdd does not seem to work well with edge weights.

Edit 2:
A concrete example of how I would go about constructing the weighted edge would be

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